Mathematical Study of the Role of Gametocytes and an Imperfect Vaccine on Malaria Transmission Dynamics

A mathematical model is developed to assess the role of gametocytes (the infectious sexual stage of the malaria parasite) in malaria transmission dynamics in a community. The model is rigorously analysed to gain insights into its dynamical features. It is shown that, in the absence of disease-induced mortality, the model has a globally-asymptotically stable disease-free equilibrium whenever a certain epidemiological threshold, known as the basic reproduction number (denoted by ℛ₀), is less than unity. Further, it has a unique endemic equilibrium if ℛ₀>1. The model is extended to incorporate an imperfect vaccine with some assumed therapeutic characteristics. Theoretical analyses of the model with vaccination show that an imperfect malaria vaccine could have negative or positive impact (in reducing disease burden) depending on whether or not a certain threshold (denoted by ∇) is less than unity. Numerical simulations of the vaccination model show that such an imperfect anti-malaria vaccine (with a modest efficacy and coverage rate) can lead to effective disease control if the reproduction threshold (denoted by ℛ_vac) of the disease is reasonably small. On the other hand, the disease cannot be effectively controlled using such a vaccine if ℛ_vac is high. Finally, it is shown that the average number of days spent in the class of infectious individuals with higher level of gametocyte is critically important to the malaria burden in the community.     

Dynamical analysis of a multi-strain model of HIV in the presence of anti-retroviral drugs

One major drawback associated with the use of anti-retroviral drugs in curtailing HIV spread in a population is the emergence and transmission of HIV strains that are resistant to these drugs. This paper presents a deterministic HIV treatment model, which incorporates a wild (drug sensitive) and a drug-resistant strain, for gaining insights into the dynamical features of the two strains, and determining effective ways to control HIV spread under this situation. Rigorous qualitative analysis of the model reveals that it has a globally asymptotically stable disease-free equilibrium whenever a certain epidemiological threshold (R t 0) is less than unity and that the disease will persist in the population when this threshold exceeds unity. Further, for the case where R t 0 > 1, it is shown that the model can have two co-existing endemic equilibria, and competitive exclusion phenomenon occurs whenever the associated reproduction number of the resistant strain (R t r) is greater than that of the wild strain (R t w). Unlike in the treatment model, it is shown that the model without treatment can have a family of infinitely many endemic equilibria when its associated epidemiological threshold (R(0)) exceeds unity. For the case when [Formula in text], it is shown that the widespread use of treatment against the wild strain can lead to its elimination from the community if the associated reduction in infectiousness of infected individuals (treated for the wild strain) does not exceed a certain threshold value (in this case, the use of treatment is expected to make R t w < R t r.     

To Cut or Not to Cut: A Modeling Approach for Assessing the Role of Male Circumcision in HIV Control

A recent randomized controlled trial shows a significant reduction in women-to-men transmission of HIV due to male circumcision. Such development calls for a rigorous mathematical study to ascertain the full impact of male circumcision in reducing HIV burden, especially in resource-poor nations where access to anti-retroviral drugs is limited. First of all, this paper presents a compartmental model for the transmission dynamics of HIV in a community where male circumcision is practiced. In addition to having a disease-free equilibrium, which is locally-asymptotically stable whenever a certain epidemiological threshold is less than unity, the model exhibits the phenomenon of backward bifurcation, where the disease-free equilibrium coexists with a stable endemic equilibrium when the threshold is less than unity. The implication of this result is that HIV may persist in the population even when the reproduction threshold is less than unity. Using partial data from South Africa, the study shows that male circumcision at 60% efficacy level can prevent up to 220,000 cases and 8,200 deaths in the country within a year. Further, it is shown that male circumcision can significantly reduce, but not eliminate, HIV burden in a community. However, disease elimination is feasible if male circumcision is combined with other interventions such as ARVs and condom use. It is shown that the combined use of male circumcision and ARVs is more effective in reducing disease burden than the combined use of male circumcision and condoms for a moderate condom compliance rate.     

Theoretical Assessment of Public Health Impact of Imperfect Prophylactic HIV-1 Vaccines with Therapeutic Benefits

This paper presents a number of deterministic models for theoretically assessing the potential impact of an imperfect prophylactic HIV-1 vaccine that has five biological modes of action, namely “take,” “degree,” “duration,” “infectiousness,” and “progression,” and can lead to increased risky behavior. The models, which are of the form of systems of nonlinear differential equations, are constructed via a progressive refinement of a basic model to incorporate more realistic features of HIV pathogenesis and epidemiology such as staged progression, differential infectivity, and HIV transmission by AIDS patients. The models are analyzed to gain insights into the qualitative features of the associated equilibria. This allows the determination of important epidemiological thresholds such as the basic reproduction numbers and a measure for vaccine impact or efficacy. The key findings of the study include the following (i) if the vaccinated reproduction number is greater than unity, each of the models considered has a locally unstable disease-free equilibrium and a unique endemic equilibrium; (ii) owing to the vaccine-induced backward bifurcation in these models, the classical epidemiological requirement of vaccinated reproduction number being less than unity does not guarantee disease elimination in these models; (iii) an imperfect vaccine will reduce HIV prevalence and mortality if the reproduction number for a wholly vaccinated population is less than the corresponding reproduction number in the absence of vaccination; (iv) the expressions for the vaccine characteristics of the refined models take the same general structure as those of the basic model.     

Mathematical Study of a Staged-Progression HIV Model with Imperfect Vaccine

A staged-progression HIV model is formulated and used to investigate the potential impact of an imperfect vaccine. The vaccine is assumed to have several desirable characteristics such as protecting against infection, causing bypass of the primary infection stage, and offering a disease-altering therapeutic effect (so that the vaccine induces reversal from the full blown AIDS stage to the asymptomatic stage). The model, which incorporates HIV transmission by individuals in the AIDS stage, is rigorously analyzed to gain insight into its qualitative features. Using a comparison theorem, the model with mass action incidence is shown to have a globally-asymptotically stable disease-free equilibrium whenever a certain threshold, known as the vaccination reproduction number, is less than unity. Furthermore, the model with mass action incidence has a unique endemic equilibrium whenever this threshold exceeds unity. Using the Li-Muldowney techniques for a reduced version of the mass action model, this endemic equilibrium is shown to be globally-asymptotically stable, under certain parameter restrictions. The epidemiological implications of these results are that an imperfect vaccine can eliminate HIV in a given community if it can reduce the reproduction number to a value less than unity, but the disease will persist otherwise. Furthermore, a future HIV vaccine that induces the bypass of primary infection amongst vaccinated individuals (who become infected) would decrease HIV prevalence, whereas a vaccine with therapeutic effect could have a positive or negative effect at the community level.     

Effect of pathogen-resistant vectors on the transmission dynamics of a vector-borne disease

A model is introduced for the transmission dynamics of a vector-borne disease with two vector strains, one wild and one pathogen-resistant; resistance comes at the cost of reduced reproductive fitness. The model, which assumes that vector reproduction can lead to the transmission or loss of resistance (reversion), is analyzed in a particular case with specified forms for the birth and force of infection functions. The vector component can have, in the absence of disease, a coexistence equilibrium where both strains survive. In the case where reversion is possible, this coexistence equilibrium is globally asymptotically stable when it exists. This equilibrium is still present in the full vector-host system, leading to a reduction of the associated reproduction number, thereby making elimination of the disease more feasible. When reversion is not possible, there can exist an additional equilibrium with only resistant vectors.     

Threshold dynamics of a non-autonomous SEIRS model with quarantine and isolation

A model for assessing the effect of periodic fluctuations on the transmission dynamics of a communicable disease, subject to quarantine (of asymptomatic cases) and isolation (of individuals with clinical symptoms of the disease), is considered. The model, which is of a form of a non-autonomous system of non-linear differential equations, is analysed qualitatively and numerically. It is shown that the disease-free solution is globally-asymptotically stable whenever the associated basic reproduction ratio of the model is less than unity, and the disease persists in the population when the reproduction ratio exceeds unity. This study shows that adding periodicity to the autonomous quarantine/isolation model developed in Safi and Gumel (Discret Contin Dyn Syst Ser B 14:209–231, 2010) does not alter the threshold dynamics of the autonomous system with respect to the elimination or persistence of the disease in the population.     

Analysis of Risk-Structured Vaccination Model for the Dynamics of Oncogenic and Warts-Causing HPV Types

A new deterministic model is designed and used to assess the community-wide impact of mass vaccination of new sexually active individuals on the dynamics of the oncogenic and warts-causing HPV types. Rigorous qualitative analyses of the model, which incorporates the two currently available anti-HPV vaccines, reveal that it undergoes competitive exclusion when the reproduction of one HPV risk type (low/high) exceeds unity, while that of the other HPV risk type is less than unity. For the case when the reproduction numbers of the two HPV risk types (low/high) exceed unity, the two risk types co-exist. It is shown that the sub-model with the low-risk HPV types only has at least one endemic equilibrium whenever the associated reproduction threshold exceeds unity. Furthermore, this sub-model undergoes a re-infection-induced backward bifurcation under certain conditions. In the absence of the re-infection of recovered individuals and cancer-induced mortality in males, the associated disease-free equilibrium of the full (risk-structured) model is shown to be globally asymptotically stable whenever the reproduction number of the model is less than unity (that is, the full model does not undergo backward bifurcation under this setting). It is shown, via numerical simulations, that the use of the Gardasil vaccine could lead to the effective control of HPV in the community if the coverage rate is in the range of 73–95 % (84 %). If 70 % of the new sexually active susceptible females are vaccinated with the Gardasil vaccine, additionally vaccinating 34–56 % (45 %) of the new sexually active susceptible males can lead to the effective community-wide control (or elimination) of the HPV types.     

Backward Bifurcation and Optimal Control in Transmission Dynamics of West Nile Virus

The paper considers a deterministic model for the transmission dynamics of West Nile virus (WNV) in the mosquito-bird-human zoonotic cycle. The model, which incorporates density-dependent contact rates between the mosquito population and the hosts (birds and humans), is rigorously analyzed using dynamical systems techniques and theories. These analyses reveal the existence of the phenomenon of backward bifurcation (where the stable disease-free equilibrium of the model co-exists with a stable endemic equilibrium when the reproduction number of the disease is less than unity) in WNV transmission dynamics. The epidemiological consequence of backward bifurcation is that the classical requirement of having the reproduction number less than unity, while necessary, is no longer sufficient for WNV elimination from the population. It is further shown that the model with constant contact rates can also exhibit this phenomenon if the WNV-induced mortality in the avian population is high enough. The model is extended to assess the impact of some anti-WNV control measures, by re-formulating the model as an optimal control problem with density-dependent demographic parameters. This entails the use of two control functions, one for mosquito-reduction strategies and the other for personal (human) protection, and redefining the demographic parameters as density-dependent rates. Appropriate optimal control methods are used to characterize the optimal levels of the two controls. Numerical simulations of the optimal control problem, using a set of reasonable parameter values, suggest that mosquito reduction controls should be emphasized ahead of personal protection measures.     

Medium-chain-length poly-3-hydroxyalkanoates-carbon nanotubes composite anode enhances the performance of microbial fuel cell

Bioprocess and Biosystems Engineering - Insufficient power generation from a microbial fuel cell (MFC) hampers its progress towards utility-scale development. Electrode modification with...